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Equations of Equilibrium In Structural analysis

A structure or one of the members considered in equilibrium when it maintains a balance of force and moment. this required satisfying force and moment equation in three axis.

                                     ΣFx=0,            ΣFy=0,               ΣFz=0
                                     ΣMx=0,           ΣMy=0,               ΣMz=0

for a structure in a single plane and since the forces are coplanar the equation of equilibrium reduced to 


where ΣFx is the sum of the component of the force in the x-axis, ΣFy is the sum of the component of the force in the y-axis, ΣMo is the algebraic sum of the moment of the component of the forces about an axis perpendicular to x and y-axis and passing through point o. whenever these equations are applied, free body diagram shall be drawn for the structure or it is members, if the analysis required for a member, this member should be isolated and all forces and moments acting on this member shall be applied, In this regard, the types of reactions at the supports can be determined using Table 1. forces common to two members will have the same magnitude and opposite direction on the free body diagram as shown in figure 1. 
Figure 1

if the internal loading at a specified point required, then cut shall be made at this point and the segment should be isolated and free body diagram will be drawn to determine internal forces, In general, the internal loadings acting at the section will consist of a normal force N, shear force V, and bending moment M, as shown in figure 2.

Figure 2 

Table 1


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